article published yes António Girão author Zach Hunter author Yuval Wigderson author 2d0023a0-1567-11f0-833d-d5c1e476d4b5 A highly influential result of Nikiforov states that if an n-vertex graph G contains at least γnh copies of a fixed h-vertex graph H, then G contains a blowup of H of order Ωγ,H(logn). While the dependence on n is optimal, the correct dependence on γ is unknown; all known proofs yield bounds that are polynomial in γ, but the best known upper bound, coming from random graphs, is only logarithmic in γ. It is a major open problem to narrow this gap. We prove that if H is triangle-free, then the logarithmic behavior of the upper bound is the truth. That is, under the assumptions above, G contains a blowup of H of order ΩH(logn/log(1/γ)). This is the first non-trivial instance where the optimal dependence in Nikiforov’s theorem is known. As a consequence, we also prove an upper bound on multicolor Ramsey numbers of blowups of triangle-free graphs, proving that the dependence on the number of colors is polynomial once the blowup is sufficiently large. This shows that, from the perspective of multicolor Ramsey numbers, blowups of fixed triangle-free graphs behave like bipartite graphs. Alliance of Diamond Open Access Journals2025 eng 2408.1291310.19086/aic.2025.10 10 yes A. Girão, Z. Hunter, Y. Wigderson, Advances in Combinatorics 10 (2025). Girão A, Hunter Z, Wigderson Y. Blowups of triangle-free graphs. <i>Advances in Combinatorics</i>. 2025;10. doi:<a href="https://doi.org/10.19086/aic.2025.10">10.19086/aic.2025.10</a> Girão, António, Zach Hunter, and Yuval Wigderson. “Blowups of Triangle-Free Graphs.” <i>Advances in Combinatorics</i>. Alliance of Diamond Open Access Journals, 2025. <a href="https://doi.org/10.19086/aic.2025.10">https://doi.org/10.19086/aic.2025.10</a>. Girão A, Hunter Z, Wigderson Y. 2025. Blowups of triangle-free graphs. Advances in Combinatorics. 10. A. Girão, Z. Hunter, and Y. Wigderson, “Blowups of triangle-free graphs,” <i>Advances in Combinatorics</i>, vol. 10. Alliance of Diamond Open Access Journals, 2025. Girão, A., Hunter, Z., &#38; Wigderson, Y. (2025). Blowups of triangle-free graphs. <i>Advances in Combinatorics</i>. Alliance of Diamond Open Access Journals. <a href="https://doi.org/10.19086/aic.2025.10">https://doi.org/10.19086/aic.2025.10</a> Girão, António, et al. “Blowups of Triangle-Free Graphs.” <i>Advances in Combinatorics</i>, vol. 10, Alliance of Diamond Open Access Journals, 2025, doi:<a href="https://doi.org/10.19086/aic.2025.10">10.19086/aic.2025.10</a>. 221722026-06-29T10:56:44Z2026-07-14T08:50:55Z